Bound vortex light in an emulated topological defect in photonic lattices

Topology have prevailed in a variety of branches of physics. And topological defects in cosmology are speculated akin to dislocation or disclination in solids or liquid crystals. With the development of classical and quantum simulation, such speculative topological defects are well-emulated in a variety of condensed matter systems. Especially, the underlying theoretical foundations can be extensively applied to realize novel optical applications. Here, with the aid of transformation optics, we experimentally demonstrated bound vortex light on optical chips by simulating gauge fields of topological linear defects in cosmology through position-dependent coupling coefficients in a deformed photonic graphene. Furthermore, these types of photonic lattices inspired by topological linear defects can simultaneously generate and transport optical vortices, and even can control the orbital angular momentum of photons on integrated optical chips.


I) The Dirac equation under a gauge field of the cosmic string
We start by considering the line element of a spacetime metric for a static cylindrically symmetric cosmic string at a certain plane in the low-mass limit with straight and infinite long length: 2 = − 2 + 2 + 2 2 2 (S1) We obtain the element of metric as = −1, = 1, = 2 2 , = −1, = 1, = 1 ( 2 2 ) ⁄ ( 2) Then we can get the calculated Christoffel symbol as The Einstein convention is adopted here where the repeated indices are summed over.
We have for nonzero Christoffel symbol as The calculated spinor connection can be directly obtained as Owing to ∇ = + Ω ， = , Eq.(S9) can be written as: We choose 0 = , 1 = , 2 = − ,then , we obtain the Hamilton as: The equation (S13) describes the Dirac Fermion with anisotropic velocity in polar coordinates as ⁄ = , and the magnetic field under the presence of the cosmic string as: sin 2 cos 2 0 | = −2 ( sin 2 + cos 2 ) ⃑ = − 2 cos 3 ⃑ (S14) To find the solution of Eq.(S13), we take the ansatz = ( ) − and obtain the equations: Although there is not existing general analytical solution for Eq.(S15), we can achieve special solution after some approximations: 1) When considering → 0, Eq.(S15) can be written as: 2) At the Dirac point = 0, the coupled equations of Eq.(S15) become decoupled and can be written as: we obtain the analytic solution as: The radiation mode according to theoretical calculation respectively based on Eq.(S19) and the deformed optical graphene lattice just as shown in Fig. S1 are good agreement. The radiating mode in the deformed optical graphene lattice.

II) The gauge field constructed by a deformed photonic graphene
We are interested in the low-energy Hamiltonian, i.e, the effective Dirac Hamiltonian, of a honeycomb lattice with anisotropy in the nearest-neighbor hopping parameters. As unstrained graphene, our lattice consists of a triangular Bravais lattice with a pair of atoms (open and filled circles in Fig. S2(a)) located in its primitive cell.
However, we consider that the hopping between nearest sites are space dependent and, in general, are characterized by three hopping parameters 1 , 2 , 3 . Within this nearestneighbor tight-binding model, one can demonstrate that the Hamiltonian in momentum space can be represented by a 2×2 matrix of the form where 1 , 2 , 3 are the nearest-neighbor vectors, and are respectively defined as 1 = (1,0) , 2 = 2 (−1, √3) , 3 = 2 (−1, −√3) . From Eq.(S20) follows that the dispersion relation is given by two bands, As is well document, for the isotropic case 1 = 2 = 3 = 0 , the Dirac points ⃗ ⃗ , which are determined by condition ( ⃗ ⃗ ) = 0, coincide with the corners of the first Brillouin zone. Then, to obtain the Dirac Hamiltonian in this case, one can simply expand the Hamiltonian (S20) around a corner, e.g. 0 = (0, 4 (3√3 ) ⁄ ) (just as shown in Fig. S2(b)). However, for the considered anisotropic case 1 = 0 (1 + ∆ ), the Dirac point do not coincide with the corners of the first Brillouin zone as shown in Fig. S1(c). The condition ( ⃗ ⃗ ) = 0, can be equivalently rewritten as Then, one can propose the position of ⃗ ⃗ in the form where the unknown shift will be looked for as a linear combination on the Thus, one obtains that the shift is given by Furthermore, the effective Dirac Hamiltonian around Dirac point ⃗ ⃗ by considering momenta close to the Dirac point ⃗ ⃗ , i.e., ⃗ = ⃗ ⃗ + , and expanding to first order and {∆ }, Hamiltonian (S20) transform as Collecting the contribution of each term in this expression, one obtains: (2 2 + 2 3 − 1 ) ) . Then, considering the contribution of each term in Eq.(S26), the effective Dirac Hamiltonian around ⃗⃗⃗ has the form: In order to obtain Dirac fermion with anisotropic velocity in polar coordinates, one can take perturbed hopping parameters as 1 = 2 , 2 = − (2 − 3 ) , 3 = − (2 + 3 ), and obtain effective Dirac Hamiltonian with ( ) = ( − ) being rotate operator. Based on effective Dirac Hamiltonian, the anisotropic velocity of Dirac fermion in polar is taken as = (1− ) = . One can also obtain the gauge field as = 2 , = 2 , which means that the momentum shift of Dirac cone ( ⃗ ⃗ ) circles around the unperturbed Dirac cone ( ⃗ ⃗ 0 ) (just as shown in Fig. S2(c)). Finally, the quantum field under the gauge field in the presence of cosmic strings are achievable.
Due to the similarities of the two Hamiltonians with different valley index, the evolution of different valleys has the same behavior. To study the qualitative features of the photonic lattice to emulate cosmic string, we calculate the energy spectrum of the corresponding tight-binding models. Fig. S4(a) shows the deformed photonic graphene lattice with a position-dependent coefficient to emulate the cosmic string with a gauge field. We convert the lattices into tight-binding Hamiltonians with nearest neighbor hopping as 1 = 0 (1 + 2 ), 2 = 0 (1 − (2 − 3 )) , 3 = 0 (1 − (2 + 3 )) (Here 0 = 1 is chosen for convenience, not to fit experiments). Fig. S4(b) shows energy spectrum of the deformed lattice with = − 1 3 ⁄ , which indicates the cosmic string with the density parameter = 1 2 ⁄ . After numerical calculations, we find there exist two pairs of special degenerate energies = ± 0 as shown in the inset of Fig. S4(b). The eigen state number = 210 ( = 211 ) and = 391 ( = 390) , which respectively correspond to the energy spectrum of 0 and − 0 , have same intensity distribution.
And the most energy is confined to the innermost layer of the deformed lattices as shown in Fig. S4(c) and Fig. S4(d). At the same time, from the phase distribution of the main intensity of the innermost layer sites, the eigen state number = 210 with = − 0 is a symmetric mode, while the eigen state number = 390 with = 0 is an antisymmetric mode.
IV) The evolution of vortex state around the cosmic string. Furthermore, if we excite all the sites of the innermost layer of such a lattice with an angular momentum (here we also take = 1 as an example), we find that although some energy spreads out, most energy are confined in the innermost layer with the same phase difference 1 = 6 ⁄ , just as shown in Fig. S6(e) and (f). When the light propagates some enough distance, the intensity of the same type is constant during propagation, while there is a little discrepancy between the different type A (site 1,3,5) and type B site (site 2,4,6). Nevertheless, the vortex light can be efficiently bound around the origin of the deformed lattice to emulate cosmic string.

V) The discussion between the curvature of cosmic strings and the exist of bound vortices
Considering the line element of cosmic-string space time as Eq.S1, the corresponding Riemann curvature is given by  To qualitatively analyze the relation about the existence of a bound vortex mode with the density parameter , we can compare the coupling coefficient in the innermost ring layer with that between the innermost layer and the adjacent out layer (see Fig.   S7(d)). The underlie reason is that for such a bound vortex mode, most energy are confined in the innermost ring layer. After considering the designed coupling coefficient, we find that the waveguides between the innermost ring layer have the same coupling coefficient dubbed as , while for the case of the innermost ring layer between adjacent out layer they also have the same coupling coefficient dubbed as .
When the density parameter = 1 leading to = , there obviously exist no bound vortex light. For the case of < 1, we find that this result of > , which means that photons tend to couple among these waveguides in the innermost ring layer.
Whereas for the case of > 1, it has the corresponding result of < , resulting in that photons tend to escape from the innermost ring layer.
VI) The discussion on the details in experiment Fig. S8 (a) The relationship between the coupling coefficients and the separation between adjacent waveguides, the inset shown the distribution evolution for waveguide separated 15 μm; (b) The coincidence counts for indistinguishable photons after a 3D 2×2 photonic beam splitter.
We fabricate the designed photonic lattice using the femtosecond laser direct writing technology. According to the characterized relationship between the coupling coefficients and the separation between adjacent waveguides, as Fig. S8(a) shown, we fabricate the samples in borosilicate glass substrate (refractive index n0=1.514 for the writing laser) using the femtosecond laser system operating at a wavelength of 513 nm, a repetition rate of 1 MHz and a pulse duration of 290 fs. The light is reshaped with a cylindrical lens and then is focused inside the sample with a 50× microscope objective (NA=0.55). The substrates are continuously moved using a high-precision three-axis translation stage with a constant velocity of 10 mm/s, and the lattices are created by the laser-induced refractive index increase.
In experiment, we firstly characterize the coupling between two adjacent waveguides to calculate the coupling strength and demonstrate the stabilization of coupling process. As shown in Fig. S8(a), we inject the photon into one of the adjacent two waveguides, and measure dynamical intensity of photon in the excited waveguide, the measured result is cosine oscillation as shown in inset. From the photon dynamics between the adjacent waveguides, we can obtain the coupling strength, which depends on the separation space between the waveguides. To show the stabilization of coupling process, we implement the HOM interference for two indistinguishable photons after a 3D 2×2 photonic beam splitter (where the waveguides are not straight like in Fig. 3(e) of the main text). The HOM dip is successfully observed in the result shown in Fig.   S8(b), which means the phase and polarization of photon are less influenced. Apart from the imperfections, loss is also the point that should be concerned in experiments. In our experiment, though the laser-written waveguides are lossy, such lossy (0.2-0.3dB/cm in our experiment) is uniform for all the sites in the lattice, which will not influent the dynamics of photons in the deformed photonic graphene, especially we excite lattice with coherent photons.
In the process of measuring the photonic chip, the light is injected into lattice using a 20× objective lens, and the outgoing probability distributions of the photons outgoing from the chip are observed using a 10× microscope objective lens and a CCD camera, and the results are recorded in both images and corresponding raw digital data. The figures showing photon distributions in the main text, such as Fig. 2(b,d,g) and Fig.   3(d), are raw images after adjusting the color bar for appropriate visual results. The analyzed results, such as the diffusion size in Fig. 2(e) and bound index in Fig. 2(h), are obtained from the corresponding raw digital data.
VII) The experiment design of photonic lattice. Fig. S10 (a) The ideal photonic graphene with an identical lattice distance but anisotropic coupling coefficients 1,2,3 . The symbol is the origin center of the photonic graphene where the emulated topological linear defect locates.
is the is azimuth angle of the waveguide site compared to the supposed location of the cosmic defect . (b) The coupling efficient 1,2,3 as a function of the azimuth angle . (c) The schematic of the deformed photonic graphene using the relationship between the coupling coefficients and the separation distance between adjacent waveguides. (d) The experimental deformed photonic graphene.
Considering the density parameter = 1 2 ⁄ , the controlling parameter in the coupling coefficient is = − 1 3 ⁄ . In experiment, we take 0 = 0.158. And the characterized relationship between the coupling coefficients and the separation distance between adjacent waveguides satisfies the formula as = −log( 3.30 ⁄ ) 0.19 ⁄ . Fig. S10c shows the deformed photonic graphene lattice to satisfy the designed coupling coefficient required by the emulated cosmic string. And Movies S1: This video exhibits that when a single waveguide in the innermost ring layer around the origin of the deformed photonic graphene, which emulates a cosmic string, is pumped, a large portion of energy was confined and twisted around the string with a clockwise direction.
Movies S2: This video exhibits that when a single waveguide in the uniform photonic graphene is pumped, there is no bound state and light is spreading outward.
Movies S3: This video exhibits that when three same type waveguide sites in the innermost ring layer around the origin of the deformed photonic graphene, which emulates a cosmic string, are pumped with the different phases ( 1 = 0 , 2 = 2 3 ⁄ , 3 = 4 3 ⁄ ), there is a vortex bound state circling around the origin of the cosmic string.
Movies S4: This video exhibits that when three same type waveguide sites in the uniform photonic graphene are pumped with the same phase, there exist only the radiating mode.